scholarly journals On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1561 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Derivative ◽  
Initial Value Problems ◽  
Monotone Functions ◽  
Initial Value ◽  
Relaxation Equation ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Completely Monotone Functions ◽  
Completely Monotone ◽  
Fractional Relaxation

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative.

scholarly journals Fractional relaxation with time-varying coefficient

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Roberto Garra ◽  
Andrea Giusti ◽  
Francesco Mainardi ◽  
Gianni Pagnini
Keyword(s):  
Point Of View ◽  
Fractional Integrals ◽  
Time Variability ◽  
Monotone Functions ◽  
Completely Monotone Functions ◽  
Bessel Operators ◽  
Completely Monotone ◽  
Fractional Relaxation ◽  
Time Origin ◽  
Physical Phenomena

AbstractFrom the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

scholarly journals A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel

2020 ◽  
Vol 23 (1) ◽  
pp. 211-223 ◽  
Author(s):  
Andrzej Hanyga
Keyword(s):  
Fractional Derivative ◽  
Fractional Integral ◽  
Necessary Condition ◽  
Monotone Functions ◽  
Existence Of A Solution ◽  
Completely Monotone Functions ◽  
Completely Monotone ◽  
Generalized Fractional Integral ◽  
Regular Kernel ◽  
Generalized Fractional Derivative

AbstractThe problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Akbar Zada
Keyword(s):  
Fractional Derivative ◽  
Positive Solutions ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Relaxation Equation ◽  
Monotone Iterative ◽  
Banach Contraction ◽  
Uniqueness Of The Solution ◽  
Fractional Relaxation ◽  
The Banach Contraction Principle

Abstract This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

scholarly journals On the Correlation Structure of a Lévy-Driven Queue

2008 ◽  
Vol 45 (4) ◽  
pp. 940-952 ◽  
Author(s):  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Random Variable ◽  
Correlation Structure ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Complementary Distribution ◽  
Completely Monotone ◽  
Server Queue ◽  
Heavy Tailed

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined asr(t) := cov(Q0,Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show thatr(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics ofr(t), for larget, for the cases of light-tailed and heavy-tailed Lévy inputs.

scholarly journals A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Ruifeng Wu ◽  
Huilai Li ◽  
Tieru Wu
Keyword(s):  
Initial Value Problems ◽  
Bernoulli Polynomials ◽  
Interpolation Scheme ◽  
Degree Polynomial ◽  
Initial Value ◽  
Linear Combinations ◽  
Shape Preserving ◽  
Runge Kutta Method ◽  
Even Order ◽  
Derivatives Of

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the functionf(x)(x∈ℝ)to approximate the derivatives off(x), we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parametercand a nonnegative integerm. Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.’s quasi-interpolation scheme.

ON THE $ \langle \rho_j,W_j \rangle $ GENERALIZED COMPLETELY MONOTONE FUNCTIONS

2020 ◽  
Vol 54 (1 (251)) ◽  
pp. 35-43
Author(s):  
B.A. Sahakyan
Keyword(s):  
Monotone Functions ◽  
Completely Monotone Functions ◽  
Completely Monotone ◽  
Class Of Functions

We consider sequences $ {\lbrace \rho_j \rbrace}_{0}^{\infty} $ $ (\rho_0 \mathclose{=} 1, \rho_j \mathclose{\geq} 1) $, $ {\lbrace \alpha_j \rbrace}_{0}^{\infty} $ $ (\alpha_0 \mathclose{=} 1, \alpha_j \mathclose{=} 1 \mathclose{-} (1/\rho_j )) $, $ {\lbrace W_j (x) \rbrace}_{0}^{\infty} \mathclose{\in} W $, where $$ W \mathclose{=} \lbrace {\lbrace W_j (x) \rbrace}_{0}^{\infty} / W_0 (x) \mathclose{\equiv} 1, W_j (x) \mathclose{>} 0, {W}_{j}^{\prime} (x) \mathclose{\leq} 0, W_j (x) \mathclose{\in} C^\infty [0,a] \rbrace, $$ $ C^\infty [0,a] $ is the class of functions of infinitely differentiable. For such sequences we introduce systems of operators $ {\lbrace {A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $, $ {\lbrace \tilde{A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $ and functions $ {\lbrace {U}_{a,n} (x) \rbrace}_{0}^{\infty} $, $ {\lbrace {\Phi}_{n} (x,t) \rbrace}_{0}^{\infty} $. For a certain class of functions a generalization of Taylor–Maclaurin type formulae was obtained. We also introduce the concept of $ \langle \rho_j,W_j \rangle $ generalized completely monotone functions and establish a theorem on their representation.

Sign in / Sign up

Export Citation Format

Share Document